Making weak maps compose strictly Richard Garner Uppsala University - PowerPoint PPT Presentation

Making weak maps compose strictly Richard Garner Uppsala University CT 2008, Calais

Outline Motivation Weak maps of bicategories Weak maps of tricategories Weak maps of weak ω -categories (NB: Talk notes available at http://www.dpmms.cam.ac.uk/ ∼ rhgg2 )

Motivation Consider a category C with: ◮ Objects being higher-dimensional ——s; ◮ Morphisms being strict structure-preserving maps. Would like to derive C wk with: ◮ Same objects; ◮ Morphisms being weak structure-preserving maps.

Motivation Idea from homotopy theory: identify strict maps X ′ → ˜ weak maps X → Y Y with where: ◮ X ′ is a cofibrant replacement for X ; ◮ ˜ Y is a fibrant replacement for Y .

Example: Ch ( R ) Ch ( R ) , category of (positively graded) chain complexes over R . ◮ A strict map X → Y is a map of chain complexes;

Example: Ch ( R ) Ch ( R ) , category of (positively graded) chain complexes over R . ◮ A strict map X → Y is a map of chain complexes; ◮ A strict map X ′ → Y is a map which preserves the R -module structure only up to homotopy ;

Example: Ch ( R ) Ch ( R ) , category of (positively graded) chain complexes over R . ◮ A strict map X → Y is a map of chain complexes; ◮ A strict map X ′ → Y is a map which preserves the R -module structure only up to homotopy ; ◮ A strict map X → ˜ Y is a map which preserves the di ff erential only up to homotopy ;

Example: Ch ( R ) Ch ( R ) , category of (positively graded) chain complexes over R . ◮ A strict map X → Y is a map of chain complexes; ◮ A strict map X ′ → Y is a map which preserves the R -module structure only up to homotopy ; ◮ A strict map X → ˜ Y is a map which preserves the di ff erential only up to homotopy ; ◮ A strict map X ′ → ˜ Y (= weak map X → Y ) is a map which preserves the R -module structure and the di ff erential only up to homotopy .

Example: f/ Cat /Z Let f : X → Z in Cat . Can form the interval category f/ Cat /Z : g h ◮ Objects are X − → Y − → Z with hg = f ; ◮ Morphisms are commutative diamonds: X g j Y W . h k Z

Example: f/ Cat /Z Corresponding weak maps should be pseudo-commutative diamonds: X g j ∼ = Y W ∼ = h k Z

Example: f/ Cat /Z Corresponding weak maps should be pseudo-commutative diamonds: X g j ∼ = Y W ∼ = h k Z ... and these are precisely strict maps X � g ′ j Y ′ � W , h ′ � k Z where:

Example: f/ Cat /Z j k − − → W − − → Z is: ... fibrant replacement of X λ k ◦ j ρ k X − − − − − → W ↓ ∼ = k − − − → Z

Example: f/ Cat /Z j k − − → W − − → Z is: ... fibrant replacement of X λ k ◦ j ρ k X − − − − − → W ↓ ∼ = k − − − → Z g h and cofibrant replacement of X − − − → Y − − → Z is: l g h ◦ r g X − − − → g ↑ ∼ = Y − − − − − → Z .

How to compose weak maps? Idea from category theory: ◮ Cofibrant replacement should be a comonad ( – ) ′ : C → C ; ◮ Fibrant replacement should be a monad � ( – ): C → C ; X ) ′ → � ◮ There should be a distributive law d X : ( ˜ X ′ .

How to compose weak maps? Idea from category theory: ◮ Cofibrant replacement should be a comonad ( – ) ′ : C → C ; ◮ Fibrant replacement should be a monad � ( – ): C → C ; X ) ′ → � ◮ There should be a distributive law d X : ( ˜ X ′ . Now composition of weak maps is two-sided Kleisli composition : f g ( X ′ Y ) ◦ ( Y ′ → ˜ → ˜ − − − − − − Z ) := f ′ g ˜ µ Z ∆ X d Y → ˜ → � X ′ → X ′′ → (˜ Y ) ′ ˜ → ˜ Y ′ − − − − − − − − − − − − − − − − Z Z .

Example: f/ Cat /Z ◮ Cofibrant replacement is a comonad [Grandis–Tholen 2006]; ◮ Fibrant replacement is a monad [loc. cit.]; ◮ There is a distributive law between them; and corresponding Kleisli composition is what you think it is: pasting of pseudo-commutative diamonds.

In general If a (locally presentable) category C has a cofibrantly generated model structure on it, then: ◮ Cofibrant replacement can be made a comonad [G. 2008]; ◮ Fibrant replacement can be made a monad [loc. cit.]; ◮ But not clear how to get a distributive law between them! So in this talk, we focus on the case where every object is fibrant . (As then we only need cofibrant replacement comonad).

Weak maps of bicategories Consider the category Bicat s : ◮ Objects are bicategories; ◮ Morphisms are strict homomorphisms. There is a cofibrantly generated model structure on Bicat s [Lack, 2004], wherein: ◮ Weak equivalences are biequivalences; ◮ Every object is fibrant. What are the corresponding weak maps?

First we describe cofibrant replacement comonad ( – ) ′ . It is generated by the following set of maps in Bicat s : • • • • • • � , , , • • • • . • • •

Explicitly, if B is a bicategory, then B ′ is given as follows: ◮ Ignore the 2-cells and form the free bicategory FU B on the underlying 1-graph of B ;

Explicitly, if B is a bicategory, then B ′ is given as follows: ◮ Ignore the 2-cells and form the free bicategory FU B on the underlying 1-graph of B ; ◮ Factorise the counit map ǫ : FU B → B as → B ′ b a FU B − → B − where a is bijective on objects and 1-cells and b is locally fully faithful. (NB: this is the flexible replacement of [Blackwell-Kelly-Power 1989]).

Proposition (Coherence for homomorphisms) The co-Kleisli category of ( – ) ′ : Bicat s → Bicat s is isomorphic to the category Bicat of bicategories and homomorphisms.

Proposition (Coherence for homomorphisms) The co-Kleisli category of ( – ) ′ : Bicat s → Bicat s is isomorphic to the category Bicat of bicategories and homomorphisms. Proof . ◮ First define a comonad H on Bicat s such that Kl ( H ) ∼ = Bicat by construction; = ( – ) ′ as comonads. ◮ Then show that H ∼

Proposition (Coherence for homomorphisms) The co-Kleisli category of ( – ) ′ : Bicat s → Bicat s is isomorphic to the category Bicat of bicategories and homomorphisms. Proof . ◮ First define a comonad H on Bicat s such that Kl ( H ) ∼ = Bicat by construction; = ( – ) ′ as comonads. ◮ Then show that H ∼ Explicitly, given bicategory B , we form H B as follows: Start with FU B as above. Given f : X → Y in B , write [ f ]: X → Y for corresponding generator in FU B .

Now adjoin 2 -cells to FU B as follows: ◮ For each α : f ⇒ g in B , a 2 -cell [ α ]: [ f ] ⇒ [ g ] ; ◮ For each X ∈ B , a 2 -cell η X : id X ⇒ [id X ] ; f g ◮ For each X − → Y − → Z ∈ B , a 2 -cell µ g,f : [ g ] ◦ [ f ] ⇒ [ g ◦ f ] ;

Now adjoin 2 -cells to FU B as follows: ◮ For each α : f ⇒ g in B , a 2 -cell [ α ]: [ f ] ⇒ [ g ] ; ◮ For each X ∈ B , a 2 -cell η X : id X ⇒ [id X ] ; f g ◮ For each X − → Y − → Z ∈ B , a 2 -cell µ g,f : [ g ] ◦ [ f ] ⇒ [ g ◦ f ] ; And quotient out the 2 -cells by equations making: ◮ [ – ] be functorial on 2 -cells; ◮ µ g,f be natural in g and f ; ◮ The µ g,f ’s and η X ’s satisfy the unit and associativity laws. The result of this is H B .

◮ By construction, maps H B → C are in bijection with homomorphisms B → C .

◮ By construction, maps H B → C are in bijection with homomorphisms B → C . ◮ We can now make H into a comonad so that Kl ( H ) ∼ = Bicat (comonad structure on H is combinatorial essence of composition of homomorphisms—compare [Hess-Parent-Scott 2006]).

◮ By construction, maps H B → C are in bijection with homomorphisms B → C . ◮ We can now make H into a comonad so that Kl ( H ) ∼ = Bicat (comonad structure on H is combinatorial essence of composition of homomorphisms—compare [Hess-Parent-Scott 2006]). = ( – ) ′ as comonads (a normalization proof ). ◮ Finally, we show that H ∼

Weak maps of tricategories Consider the category Tricat s : ◮ Objects are tricategories; ◮ Morphisms are strict homomorphisms. We use an algebraic definition of tricategory, so Tricat s is l.f.p. and in particular cocomplete.

Weak maps of tricategories Consider the category Tricat s : ◮ Objects are tricategories; ◮ Morphisms are strict homomorphisms. We use an algebraic definition of tricategory, so Tricat s is l.f.p. and in particular cocomplete. No-one has written down the cofibrantly generated model structure on Tricat s yet, but it should have: ◮ Weak equivalences being triequivalences; ◮ Every object being fibrant. Can we describe the corresponding weak maps?

Yes: because we can describe the cofibrant replacement comonad ( – ) ′ . It’s generated by the following set of maps in Tricat s : • • • • • • • • � ; ; ; ; • • • • • • . • • •

Explicitly, if T is a tricategory, then T ′ is given as follows: ◮ Ignore the 2- and 3-cells and form the free tricategory FU T on the underlying 1-graph of T . Write ǫ : FU T → T for the counit map.